Index of Dissimilarity: A Measure of
Relative Ageing of Human Populations
Ashraf Khan Kayani,^{ }
HEC Professor of Sociology
Department of Sociology
University of the Punjab,
ZilliHuma
PhD Scholar
Department of Sociology
University of the Punjab,
Ahmad Raza
Research Fellow
Center for Organization and Development Studies
Abstract:
Index of dissimilarity is mostly used as a demographic measure of ‘evenness’ of population distributions. In the present paper, an attempt is made to see if the index can be used as a measure of relative ageing of human populations. Relative ageing is operationally defined as a change in the agedness of a closed population resulting from different population growth rates or the differences in agedness of different age distributions resulting from growth rates held constant over a long period of time.
For
the present purpose we used nine stable age distributions resulting from varying growth
rates (r = 0.0 through 4.0 with equal interval of 0.5). Using stable age distribution with
r = 2.0
(mid point) as standard, the index of dissimilarity is computed. It is observed that size
of the index does show the differences in ageing of the distributions. However, the size
alone does not show the directionality of the ageing, that is, whether a population is
younger or older. On the other hand, the size alongwith the percentage point differences
between the two distributions is a very useful measure to show the magnitude and
directionality of the agedness of human populations.
Key words: Index of dissimilarity, relative ageing, human population, agedness
INDEX OF DISSIMILARITY: A MEASURE
OF RELATIVE
AGEING
OF HUMAN POPULATIONS^{1}
Introduction
The present paper attempts to show that index of dissimilarity can be a useful measure to compare the “agedness” of the age distributions of human populations in relative terms. Expressed differently, we show the relationship between the size of the index and agedness of the age distributions. We also propose that the index alongwith the percent point difference in the distributions shows the magnitude and directionality of the agedness of human populations.
Based on the proportionate distributions and percentage point differences between them, various indices are used to measure unevenness, concentration or segregation among two or more groups (Duncan, 1957; Jakubs, 1981; Massay, 1981; White, 1986) The most commonly used index derived from proportionate distributions is called index of dissimilarity ( ? ).
The index of dissimilarity has been used by a number of demographers and social scientists (Bogue, 1969; Keyfitz, 1968, Mukherjee, 1973, and Sakoda, 1981) for different purposes. For example, Keyfitz (1968, p. 47) uses the index of dissimilarity to show how different an initial distribution of a population is from the stable distribution. Mukherjee (1973) suggests that the size of ? can be used as a criterion to determine whether a population is stable or not. Sakado (1981) generalized the ? so as to apply it to individual groups as well as to provide a single index for multiple groups.
Methods and Materials
The index of dissimilarity is calculated by taking two percentage distributions, assuming one as a standard distribution and subtracting the other distribution from the standard distribution or vice versa to get a distribution of percentage point differences. The sum of the categories of alike signs is called index of dissimilarity. One can also take the total of all percentage point differences irrespective of the signs and divide the total by 2 to get the index of dissimilarity. Arithmetically:
^{1}Revised version of an invited paper presented at 3^{rd} National Conference on Statistical,
Sciences, Islamic Countries Society of Statistical
Sciences, May 2829, 2007,

? = A_{x} _ B_{x}
2
Where A and B are two proportionate distributions and X is age category.
To achieve our objective, we have selected stable age distributions with life expectancy (e?) of 65 years and varying population growth rates for females. These stable age distributions are taken from Stable Populations Age Distributions prepared by United Nations (1990) on South Asian Pattern.
We do recognize that the stable populations are more of an abstract nature than the real ones. However, the actual age distributions for a country are generally distorted mainly due to emigration and or immigration. It is for this reason that we selected the stable age distributions which are closed to migration. Another reason for such a selection is that annual natural growth rates (r) vary from a minimum of 0.0 % to a maximum of 4.0 %.
Analysis
The selected female stable age distributions are given in table 1 with growth rates given at the top of each of the age distributions. Since growth rates decline monotonically (from 4.0 to 0.0) the nine distributions are monotonically older in the same order – an axiom of stable population theory.
Table 1. Proportionate Stable Age Distributions for Female
by Annual Growth Rates, UN South Asian Pattern
Females
E (0) = 65
Years
Percent of population in indicated age group
Growth
Rate (%): 0.0 0.5
1.0
1.5
2.0
2.5 3.0
3.5 4.0
Age group
01 14 59 1014 1519 2024 2529 3034 3539 4044 4549 5054 5559 6064 6569 7074 7579 8084 85+ 
1.46 5.57 6.86 6.82 6.80 6.76 6.72 6.68 6.62 6.54 6.43 6.26 5.98 5.54 4.90 4.01 2.93 1.81 1.31 
1.74 6.59 7.93 7.69 7.48 7.26 7.04 6.82 6.59 6.35 6.09 5.78 5.39 4.87 4.20 3.35 2.39 1.44 1.00 
2.06 7.70 9.06 8.57 8.12 7.69 7.27 6.87 6.48 6.09 5.69 5.27 4.79 4.23 3.55 2.77 1.92 1.13 0.75 
2.41 8.88 10.21 9.42 8.71 8.04 7.42 6.84 6.29 5.76 5.26 4.75 4.21 3.62 2.97 2.25 1.53 0.88 0.56 
2.78 10.13 11.38 10.25 9.24 8.32 7.48 6.73 6.03 5.39 4.80 4.22 3.65 3.06 2.45 1.82 1.20 0.67 0.41 
3.17 11.42 12.55 11.02 9.69 8.51 7.46 6.54 5.72 4.99 4.33 3.72 3.14 2.57 2.00 1.45 0.93 0.51 0.30 
3.59 12.74 13.69 11.72 10.05 8.61 7.37 6.30 5.38 4.57 3.87 3.24 2.67 2.13 1.62 1.14 0.72 0.38 0.22 
4.02 14.09 14.80 12.36 10.34 8.64 7.21 6.01 5.00 4.15 3.42 2.80 2.24 1.75 1.29 0.89 0.55 0.28 0.15 
4.46 15.45 15.87 12.82 10.54 8.59 6.99 5.69 4.62 3.73 3.00 2.39 1.87 1.42 1.03 0.69 0.41 0.21 0.11 
Total
100.0 100.0 100.0
100.0 100.0 100.0 100.0
100.0 100.0
Age distribution resulting from r=2.0 (a midpoint of rs’) is treated as the standard one. From the standard population distribution, proportions in each age group of other distributions are subtracted from the same age group in the standard population; thus getting a distribution of percentage point differences for every age group in all the nonstandard distributions. From these differences, index of dissimilarity is computed for different growth rates. The differences by age groups and the resulting ? for each of the age distributions are given in table 2.
Table 2.
Index of Dissimilarity and Percentage Point Differences from Standard Age Distribution for
other Stable Distributions, resulting from different Growth Rates
Growth
Rate (%): 0.0
0.5
1.0 1.5
2.0* 2.5 3.0 3.5
4.0
Percentage point
difference of population in indicated age group
Age group
01 14 59 1014 1519 2024 2529 3034 3539 4044 4549 5054 5559 6064 6569 7074 75+ Index of
Dissimilarity (?) 
1.32 4.56 4.52 3.43 2.44 1.56 0.76 0.05 +0.59 +1.15 +1.63 +2.04 +2.33 +2.48 +2.45 +2.19 +3.77 18.64 
1.04 3.54 3.45 2.56 1.76 1.06 0.44 +0.09 +0.56 +0.96 +1.29 +1.56 +1.74 +1.81 +1.75 +1.53 +2.55 13.85 
0.72 2.43 2.32 1.68 1.12 0.63 0.21 +0.14 +0.45 +0.7 +0.89 +1.05 +1.14 +1.17 +1.1 +0.95 +1.52 9.11 
0.37 1.25 1.17 0.83 0.53 0.28 0.06 +0.11 +0.26 +0.37 +0.46 +0.53 +0.56 +0.56 +0.52 +0.43 +0.69 4.49 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
+0.39 +1.29 +1.17 +0.77 +0.45 +0.19 0.02 0.19 0.31 0.4 0.47 0.5 0.51 0.49 0.45 0.37 0.54 4.25 
+0.81 +2.61 +2.31 +1.47 +0.81 +0.29 0.11 0.43 0.65 0.82 0.93 0.98 0.98 0.93 0.83 0.68 0.96 8.3 
+1.24 +3.96 +3.42 +2.11 +1.1 +0.32 0.27 0.72 1.03 1.24 1.38 1.42 1.41 1.31 1.16 0.93 1.3 12.16 
+1.68 +5.32 +4.49 +2.67 +1.3 +0.27 0.49 1.04 1.41 1.66 1.8 1.83 1.78 1.64 1.42 1.13 1.55 15.74 
*Standard Age Distribution
Examination of table 2 suggests that the indexes of dissimilarity for the eight nonstandard distributions show a peculiar pattern. As the growth rate increases or decreases from the standard population by the same proportion, the index sizes are almost similar; for example, the ? for growth rates 2.5% and 1.5%.
This is further clarified in Figure 1, where growth rates and the respective indexes of dissimilarity from the standard age distribution are shown in a scattergram using a linear regression model.
Discussion
The analysis suggests that the ? alone can only show the difference between the two distributions in age but does not give the direction whether the population distribution is older or younger, when compared to the standard one. However, it does indicate that the agedness of a population is changed.
To overcome the problem of directionality in ageing, assistance from distribution of percentage point differences by age groups should be of importance. The understanding of losses or gains in the proportions in the age groups leads us to argue that the index of dissimilarity (?) for two different growth levels may be the same but their distributions of gains and losses may well be entirely different (table 2). In our eight distributions of percentage point differences (of which ? is an outcome), it is evident that very high and very low growth rates have the larger values of ? and that the value of ? decreases monotonically from the highest and the lowest growth rates respectively as the value of r drifts towards the r of the standard one. This is due to the reason of treating the age distribution resulting from midlevel growth rates (r=2.0). From the growth rate r= 0.0 to r=1.5, ? increases monotonically. Similarly, it increases for distributions resulting from r=2.5 through r=4.0. For almost identical indexes of dissimilarity, the distributions of gains and losses by age groups are entirely different. For example, if we deduct the distribution with zero growth rate from the standard distribution (r = 2.0) all age groups above age 35 years gain the proportions (older populations). On the contrary, for r = 2.5 age groups above age 25 years lose the proportions (younger population).
This shows that the ? alone does not render full information and that the complete percentage
point difference distribution from which ? is derived from is
necessary to illuminate further the differences between the two distributions.
The stable age distributions in our example are the eventual effects of the constant growth rates. What the index of dissimilarity gives us, in fact, is the difference between the effects of the growth rates in the longterm. A large ? shows that the age of a population is entirely different from the initial or standard population; a small value of ? signifies little change in the age, and a medium size ? shows a moderate change in age of a population. In other words, the size of ? in any two age distributions will depict the differences in the ageing of the two populations. To see whether one population is younger or older than the other, one has to look at the percentage point differences in the age groups between the two distributions.
Source: United Nations, (1990).
Stable Population Age Distributions. Department of
International Economic and Social Affairs,
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